Decay Kinetics
CHEM 702: Lecture 6Radioactive Decay KineticsPart
1OutlineReadings: Modern Nuclear Chemistry Chapter 3; Nuclear and
Radiochemistry Chapters 4 and 5Radioactive decay kineticsBasic
decay equationsUtilization of equationsMixtures
EquilibriumBranchingNatural radiationDating
6-#Introduction to Radioactive DecayNumber of radioactive nuclei
that decay in radioactive sample decreases with timeExponential
decreaseIndependent of P, T, mass action and 1st orderConditions
associated with chemical kineticsElectron capture and internal
conversion can be affected by conditionsSpecific for isotope and
irreversibleDecay of given radionuclide is randomDecay rate
proportional to amount of parent isotoperate of decay=decay
constant*# radioactive nucleiA= lNDecay constant is average decay
probability per nucleus for a unit timeRepresented by l
6-#Basic decay equationsProbability of disintegration for a
given radioactive atom in a specific time interval is independent
past history and present circumstancesProbability of disintegration
depends only on length of time intervalProbability of decay:
p=DtProbability of not decaying: 1-p=1- Dt(1- Dt)n=probability that
atom will survive n intervals of DtnDt=t, therefore (1- Dt)n =(1-
t/n)n limn(1+x/n)n=ex, (1-t/n)n=e-t is limiting valueConsidering No
initial atomsfraction remaining unchanged after time t is N/No= e-t
N is number of atoms remaining at time t
N=Noe-t
6-#Radioactivity as Statistical Phenomenon:Binomial
DistributionRadioactive decay a random processNumber of atoms in a
given sample that will decay in a given Dt can differNeglecting
same Dt over large time differences, where the time difference is
on the order of a half lifeRelatively small Dt in close time
proximity Binomial Distribution for Radioactive
DisintegrationsReasonable model to describe decay processBin
counts, measure number of occurrences counts fall in bin numberCan
be used as a basis to model radioactive caseClassic description of
binomial distribution by coin flipProbability P(x) of obtaining x
disintegrations in bin during time t, with t short compared to
t1/2n: number of trialsp: probability of event in bin
6-#Radioactivity as Statistical Phenomenon: Error from
CountingFor radioactive disintegrationProbability of atom not
decaying in time t, 1-p, is (N/No)=e-tp=1-e-tN is number of atoms
that survive in time interval t and No is initial number of
atomsTime Intervals between DisintegrationsDistribution of time
intervals between disintegrationst and t+d Write as P(t)dt
6-#Decay StatisticsAverage disintegration rateAverage value for
a set of numbers that obey binomial distributionUse n rather than
No, replace x (probability) with r (disintegrations)
Average value for r
Solve using binomial expansion
Then differentiate with respect to x
6-#Decay StatisticsLet x=1
Related to number and probability For radioactive decay n is No
and p is (1-e-lt)Use average number of atoms disintegrating in time
tM=average number of atoms disintegrating in time tCan be measured
as counts on detectorM=No(1-e-t)For small t, M=Not Disintegration
rate is M per unit time R=M/t=NoSmall t means count time is short
compared to half lifeCorresponds to -dN/dt=N=A
6-#Decay StatisticsExpected Standard Deviation Base on expected
standard deviation from binomial distributionUse binomial expansion
and differentiate with respect to x
x=1 and p+(p-1)= 1
Variation defined asCombine
From bottom of slide 3-6
6-#Expected Standard DeviationSolve with:
Apply to radioactive decayM is the number of atoms
decayingNumber of counts for a detector
Relative error = s-1 What is a reasonable number of countsMore
counts, lower error
Countserror %
error103.1631.6210010.0010.00100031.623.1610000100.001.00
6-#Measured ActivityActivity (A) determined from measured counts
by correcting for geometry and efficiency of detectorNot every
decay is observedConvert counts to decayA=
lNA=Aoe-tUnitsCurie3.7E10 decay/s1 g 226RaA= lNBecquerel1
decay/s
6-#Half Life and Decay ConstantHalf-life is time needed to
decrease nuclides by 50%Relationship between t1/2 and
lN/No=1/2=e-tln(1/2)=-t1/2ln 2= t1/2t1/2=(ln 2)/
Large variation in half-lives for different isotopesShort
half-lives can be measuredEvaluate activity over timeObservation on
order of half-lifeLong half-lives Based on decay rate and
sampleNeed to know total amount of nuclide in sampleA=lN
6-#Exponential DecayAverage Life () for a radionuclidefound from
sum of times of existence of all atoms divided by initial number of
nuclei
1/l=1/(ln2/t1/2)=1.443t1/2=tAverage life greater than half life
by factor of 1/0.693During time 1/ activity reduced to 1/e its
initial valueTotal number of nuclei that decay over timeDoseAtom at
a time
6-#Gamma decay and Mossbauer SpectroscopyCouple with Heisenberg
uncertainty principleDE Dth/2pDt is t, with energy in
eVDE(4.133E-15 eV s/2p)/t= GG is decay widthResonance energy
G(eV)=4.56E-16/t1/2 t1/2 in secondst1/2=1 sec, t=1.44 sNeed very
short half-lives for large widthsUseful in Moessbauer
spectroscopyAbsorption distribution is centered around Eg+DE
emission centered Eg-DE . overlapping part of the peaks can be
changed by changing temperature of source and/or absorber Doppler
effect and decay width result in energy distribution near ErDoppler
from vibration of source or sample
6-#Important Equations!Nt=Noe-ltN=number of nuclei, l= decay
constant, t=timeAlso works for A (activity) or C (counts)At=Aoe-lt,
Ct=Coe-ltA= lN1/l=1/(ln2/t1/2)=1.443t1/2=t
ErrorM is number of counts
6-#Half-life calculationUsing Nt=Noe-ltFor an isotope the
initial count rate was 890 Bq. After 180 minutes the count rate was
found to be 750 BqWhat is the half-life of the
isotope750=890exp(-l*180 min)750/890=exp(-l*180 min)ln(750/890)=
-l*180 min-0.171/180 min= -l9.5E-4 min-1
=l=ln2/t1/2t1/2=ln2/9.5E-4=729.6 min
6-#Half-life calculationA=lNA 0.150 g sample of 248Cm has a
alpha activity of 0.636 mCi.What is the half-life of 248Cm?Find
A0.636 E-3 Ci (3.7E10 Bq/Ci)=2.35E7 BqFind N0.150 g x 1 mole/248 g
x 6.02E23/mole= 3.64E20 atomsl=A/N= 2.35E7 Bq/3.64E20
atoms=6.46E-14 s-1t1/2=ln2/l=0.693/6.46E-14 s-1=1.07E13 s1.07E13
s=1.79E11 min=2.99E9 h=1.24E8 d =3.4E5 a
6-#CountingA=lNYour gamma detector efficiency at 59 keV is 15.5
%. What is the expected gamma counts from 75 micromole of
241Am?Gamma branch is 35.9 % for 241AmC=(0.155)(0.359)lNt1/2=432.7
a* (3.16E7 s/a)=1.37E10 sl=ln2/1.37E10 s=5.08E-11 s-1N=75E-6 moles
*6.02E23/mole=4.52E19 atomsC=(0.155)(0.359)5.08E-11 s-1*4.52E19
=1.28E8 counts/second
6-#Decay Scheme
6-#Specific activityActivity of a given amount of radionuclide
Use A=lNUse of carrier should be includedSA of 226Ra1 g 226Ra,
t1/2= 1599 a1 g * 1 mole/226 g * 6.02E23 atoms/mole = 2.66E21 atom
= Nt1/2=1599 a *3.16E7 s/a = 5.05E10 sl=ln2/ 5.05E10 s =1.37E-11
s-1A= 1.37E-11 s-1 * 2.66E21=3.7E10 BqDefinition of a Curie!
6-#Specific Activity1 g 244Cm, t1/2=18.1 a1 g * 1 mole/244 g *
6.02E23 atoms/mole = 2.47E21 atom = Nt1/2=18.1 a *3.16E7 s/a =
5.72E8 sl=ln2/ 5.72E8 s =1.21E-9 s-1A= 1.21E-9 s-1 *
2.47E21=2.99E12 BqGeneralized equation for 1 g6.02E23/Isotope mass
*2.19E-8/ t1/2 (a)1.32E16/(Isotope mass* t1/2 (a))
Isotopet 1/2 (a)SA
(Bq/g)14C57151.65E+11228Th1.91E+003.03E+13232Th1.40E+104.06E+03233U1.59E+053.56E+08235U7.04E+087.98E+04238U4.47E+091.24E+04237Np2.14E+062.60E+07238Pu8.77E+016.32E+11239Pu2.40E+042.30E+09242Pu3.75E+051.45E+08244Pu8.00E+076.76E+05241Am4.33E+021.27E+11243Am7.37E+037.37E+09244Cm1.81E+012.99E+12248Cm3.48E+051.53E+08
6-#Specific ActivityActivity/moleN=6.02E23SA (Bq/mole) of 129I,
t1/2=1.57E7 at1/2=1.57E7 a *3.16E7 s/a = 4.96E14 sl=ln2/ 4.96E14 s
=1.397E-15 s-1A= 1.397E-15 s-1 *6.02E23=8.41E8 BqGeneralized
equationSA (Bq/mole)=1.32E16/t1/2 (a)
6-#Specific activity with carrier1E6 Bq of 152Eu is added to 1
mmole Eu.Specific activity of Eu (Bq/g)Need to find g Eu1E-3 mole
*151.96 g/mole = 1.52E-1 g=1E6 Bq/1.52E-1 g =6.58E6 Bq/g=1E9
Bq/moleWhat is SA after 5 yearst1/2=13.54 a=
6.58E6*exp((-ln2/13.54)*5)=5.09E6 Bq/g
6-#LifetimeAtom at a time chemistry261Rf lifetimeFind the
lifetime for an atom of 261Rft1/2 = 65 st=1.443t1/2t=93 sDetermines
time for experimentMethod for determining half-life
6-#Mixtures of radionuclidesComposite decaySum of all decay
particlesNot distinguished by energyMixtures of Independently
Decaying Activitiesif two radioactive species mixed together,
observed total activity is sum of two separate activities:
At=A1+A2=1N1+2N2any complex decay curve may be analyzed into its
componentsGraphic analysis of data is possible
l=0.554 hr-1t1/2=1.25 hr
l=0.067 hr-1t1/2=10.4 hr
6-#